Reservoir simulation is used in the oil and gas production industry to assess the most profitable means to locate and operate wells (boreholes) to produce subsurface accumulations of oil and gas to the surface, where the fluids can be transported, refined, and sold.
Reservoir simulation is of great interest because it infers the behavior of a real hydrocarbon-bearing reservoir from the performance of a mathematical or numerical model of that reservoir. The typical objective of reservoir simulation is to understand the complex chemical, physical, and fluid flow processes occurring in the reservoir sufficiently well to predict future behavior of the reservoir to maximize hydrocarbon recovery. Reservoir simulation calculations in such hydrocarbon systems are based on fluid flow through the reservoir being simulated. These calculations are performed with varying degrees of rigor, depending on the requirements of the particular simulation study and the capabilities of the simulation software being used.
Computer programs that use numerical simulation are used first to build a reservoir simulation model that characterizes rock and fluid properties and then to calculate the evolution of the system over time in response to planned well operations to remove saleable fluids and in some cases to replace these with less valuable fluids to maintain pressure.
The principle of numerical simulation is to numerically solve equations describing a physical phenomenon using a computer. Such equations are generally systems of ordinary differential equations and partial differential equations. As a means for numerically solving such equations, there are known the finite element method, the finite difference method, the finite volume method, and the like. Regardless of which method is used, the physical system to be modeled is divided into cells (a set of which is called a grid or mesh), and the state variables that vary in space throughout the model are represented by sets of values for each cell. A reservoir simulation grid may be built by subdividing (discretizing or gridding) the volume of interest into a large number of polyhedral cells. The number of cells commonly ranges from a few tens of thousands to several hundreds of thousands. The volume of interest is defined areally and vertically by the extent of the oil and gas accumulation and of the water that is in pressure communication with the oil and gas. The area may be several square miles, and the thickness may be hundreds or even thousands of feet. The reservoir rock properties such as porosity and permeability are typically assumed to be constant inside a cell. Other variables such as fluid pressure and phase saturation are specified at specified points, sometimes called nodes, within the cell. A link between two nodes is called a “connection.” Fluid flow between two cells is typically modeled as flow along the connection between them.
The state of a simulation cell may be defined by its pressure and its contents, i.e., the amounts of oil, gas, and water within the cell. The goal of simulation is to calculate the evolution through time of the states of all cells. This evolution is governed by the initial states and by the time-dependent removal of fluid from (production) or addition of fluid to (injection) the system by way of wells.
The state of a cell changes in time because of fluid flow between pairs of neighboring cells or between a cell and a well. Fluid flows from high pressure to low pressure. Pressure gradients are induced by removing fluid from the reservoir (production) or adding fluid to the reservoir (injection) by way of wellbores that penetrate the porous and permeable rock. Within the reservoir, fluid converges on (flows toward) producing wellbores and diverges from (flows away from) injecting wellbores.
For purposes of modeling fluid flow, approximate versions of the relevant equations are written for each cell to express the conservation of mass and the relationship between phase flow rate and pressure gradient. The simultaneous (approximate) solution of these equations for the entire collection of cells yields the pressure and contents of each cell at a single time. The equations are solved for a sequence of times to characterize the evolution of the state of the reservoir.
The properties of cells and connections play a role in determining how the reservoir simulation model performs. These properties may be derived from a geologic reservoir characterization through processing techniques used in the art.
Conventional wisdom holds that the geologic modeling process yields a highly realistic portrayal of the spatial distribution of rock properties, and that converting such a geologic model to a reservoir simulation grid should naturally yield a realistic portrayal of connectivity. But there are factors that confound this favorable outcome. Because the geologic modeling process lacks tools to explicitly visualize connectivity, the geologic model may itself contain connectivity anomalies that escape detection using current technology.
Reservoirs are commonly segmented or compartmentalized by virtue of reduced permeability at fault surfaces and at stratigraphic sequence boundaries. These surfaces are often targeted for transmissibility adjustment and are just as often the source of inconsistent or erroneous adjustments that yield connectivity anomalies.
There are four current approaches to assessing connectivity in numerical simulation. The first method is finite difference simulation. In this approach, the user runs a finite-difference reservoir simulation program and interrogates the results. This approach is slow (hours to tens of hours) and only informs the user about parts of the model that are well connected to wells. Within finite difference simulation, there are three methods of interrogating results to assess connectivity.
The first finite difference method for assessing connectivity is to visualize pressure results. This is the most common style of visual interrogation. Changes in pressure are associated with the flow of fluid. A common observation is that pressure depletion is observed in compartments previously thought to be isolated from (i.e., poorly connected to) producing wells. In such cases, the pressure field appears diffuse and does not offer any guidance as to the path that connects the pressure-depleted compartment to a producing well.
The second finite difference method for assessing connectivity is to visualize tracer-to-component-ratio results. Reservoir simulation programs may offer the ability to associate named tracers with fluid components, e.g., water. By assigning different tracer:component ratios to resident fluids and injected fluids, the user can monitor the tracer:component ratio in the reservoir model over time to get some sense of where the injected fluid goes. Similarly to pressure results, tracer-ratio results are diffuse rather than specific. The fuzziness is exacerbated by the common situation that often only infrequent snapshots of tracer:component ratio results can be afforded. A low-frequency archive yields inadequately detailed information.
The third finite difference method for assessing connectivity is to visualize flows. Customized software tools, not available in standard software, make it possible to visualize statistics of internal flows. This method is still burdened by the need to actually conduct flow simulation and by the constraint of using actual wells. It works best for injector-producer pairs that are amenable to visualizing high flow rates.
The second approach to assessing connectivity in numerical simulation uses mental tomography. In this approach the user builds a mental image of the permeability field based on viewing numerous slices through the model, where each slice displays the value of permeability using color. This technique may have been adequate for the coarsely gridded, nearly homogeneous, structured simulation grids that were built 20 years ago, but it is impractical for complex, unstructured, modern models.
The third approach to assessing connectivity in numerical simulation uses streamline simulation. In this approach, the user runs a streamline flow simulation program and interrogates the results. This approach generally shows where fluid flows. However, it requires the a priori designation of source and/or sink wells, and it is slower than embodiments of the invention described herein to yield results.
The fourth approach to assessing connectivity in numerical simulation uses geobody visualization. Customized software tools, not available in standard software, make it possible to define and visualize geobodies with respect to user-defined criteria. In graph theory terminology, a geobody is a connected component, which means for all possible pairs of cells that belong to the geobody, a path (sequence of connections) exists to connect the pair.
User input for the “define geobodies” action consists of separate criteria for nodes and connections. The criteria are logical expressions (with a pass/fail outcome) that may range from simple to complex. The criteria are evaluated using node and connection properties. If a node fails the node criteria, the node is excluded from all geobodies, and the connections in which the failed node participates are treated as failed connections. If a connection fails the connection criteria, it is excluded from paths that may connect passing nodes. A node that participates in a failed connection is not marked as a failed node, because such a node may participate in other passing connections.
The results of a “define geobodies” action are that each node is assigned values for two properties: a geobody membership index that identifies which geobody (connected component) the node belongs to and a geobody score. The geobody score is an aggregate measure of the size or importance of the geobody, such as the number of nodes or total pore volume of the geobody. The geobody membership index is commonly an integer rank, assigned in order of descending score, i.e., the highest scoring geobody is 1, the next lower scoring geobody is 2, etc. A special membership-index value such as zero or −1 marks failed nodes that belong to no geobodies. Connectivity among the failed nodes is not assessed.
The ability of “define geobodies” to reveal information is sensitive to the node and connection criteria. Good (informative) criteria yield several high-scoring geobodies and many low-scoring geobodies. The user would normally use the 3D viewer to examine each high-scoring geobody, with special interest in its three-dimensional shape. Poor (uninformative) criteria yield either a single geobody or only low-scoring geobodies. Unfortunately, trial and error are required to find good criteria, and each trial is time-consuming.
Hirsch and Schuette propose applying graph theory algorithms to rank the anticipated flow performance of different geologic model realizations and to aid in delineating contiguous regions of similar character in petroleum reservoir flow simulations. (“Graph theory applications to continuity and ranking in geologic models”, Computers & Geosciences 25, 127-139 (1999)). They report results from using three different types of graph theory algorithms in a test problem: maximum-flow, shortest-path and connected-components algorithms. Their discussion of the shortest-path method focuses on using the single-source, specific shortest path as the ranking measure, i.e., what is called the single-pair problem in the invention description to follow.
Current geologic modeling and reservoir simulation software applications lack the means to assess connectivity in a way that is fast, quantitative, and visual. Additionally, connectivity is defined poorly in existing methodologies. Connectivity may simply be determined in an yes/no methodology. A richer assessment of connectivity would identify the best path that connects Cell A with Cell B. Additional background can be found in Dijkstra, E. W., A note on two problems in connection with graphs, Numerische Mathematik, 1:269-271 (1959) and in Cormen, T. H. et al., Introduction to Algorithms, The MIT Press (1990).